P can build a wall in the same time in which Q and R together can do it. If P and Q together could do it in 30 days and R alone in 40 days, in what time could Q alone do it?

R alone can complete the wall in 40 days.

P and Q together can complete it in 30 days.

P alone takes the same time as Q and R working together.

If someone completes a job in $n$ days, their work rate is $\frac{1}{n}$ of the job per day.

Q's rate $= \frac{1}{q}$ (where $q$ is the days Q needs)

R's rate $= \frac{1}{40}$ per day

P's rate $= \frac{1}{p}$ (where $p$ is the days P needs)

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When people work together, their rates are added.

P and Q together take 30 days:

$\frac{1}{p} + \frac{1}{q} = \frac{1}{30}$ ... (Equation 1)

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P alone equals Q and R together:

$\frac{1}{p} = \frac{1}{q} + \frac{1}{40}$ ... (Equation 2)

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Substituting Equation 2 into Equation 1:

$\left(\frac{1}{q} + \frac{1}{40}\right) + \frac{1}{q} = \frac{1}{30}$

$\frac{2}{q} + \frac{1}{40} = \frac{1}{30}$

$\frac{2}{q} = \frac{1}{30} - \frac{1}{40}$

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Finding common denominator (LCM of 30 and 40 is 120):

$\frac{1}{30} = \frac{4}{120}$

$\frac{1}{40} = \frac{3}{120}$

$\frac{2}{q} = \frac{4}{120} - \frac{3}{120}$

$\frac{2}{q} = \frac{1}{120}$

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$\frac{2}{q} = \frac{1}{120}$

$q = 2 \times 120$

$q = 240$

Therefore, Q alone can build the wall in 240 days.